Problem: A parabola and an ellipse share a focus, and the directrix of the parabola is the line containing the minor axis of the ellipse. The parabola and ellipse intersect at two points. Given that the equation of the ellipse is $\frac{x^2}{25} + \frac{y^2}{9} = 1,$ find the distance between those two points.
The lengths of the semi-major and semi-minor axis are $\sqrt{25} = 5$ and $\sqrt{9} = 3.$ Then the distance from the center $(0,0)$ of the ellipse to each focus is $\sqrt{5^2-3^2} = 4,$ so the foci have coordinates $(\pm4, 0).$

Without loss of generality, assume that the parabola has its focus at $(4,0).$ Its directrix is the line containing the minor axis, which is the $y-$axis. Then the vertex of the parabola must be the point $(2,0),$ so its equation is of the form \[x = Ay^2 + 2\]for some value of $A.$ Since the distance from the vertex to the focus is $2,$ we have $2 = \tfrac{1}{4A},$ so $A = \tfrac{1}{8},$ and the equation of the parabola is \[x = \frac{y^2}8 + 2.\]The parabola and ellipse are shown together below. [asy]
size(6cm);
draw(scale(5,3)*unitcircle);
real y(real x) { return (8*x-16)**0.5; }
real z(real x) { return -y(x); }
draw(graph(y, 2, 4.5),EndArrow);
draw(graph(z, 2, 4.5),EndArrow);
dot((4,0) ^^ (-4,0));
dot((2,0));
dot((25/9,2*sqrt(14)/3) ^^ (25/9,-2*sqrt(14)/3));
draw((-7,0)--(7,0),EndArrow);
draw((0,-5)--(0,5),EndArrow);
label("$x$",(7,0),E);
label("$y$",(0,5),N);
for (int i=-6; i<=6; ++i)
	draw((i,-.2)--(i,.2));
for (int i=-4; i<=4; ++i)
	draw((-.2,i)--(.2,i));
[/asy] To find the intersection points of the parabola and ellipse, we solve the system \[\begin{aligned} \frac{x^2}{25} + \frac{y^2}9 &= 1, \\ x &=\frac{y^2}8+ 2 .\end{aligned}\]Multiplying the first equation by $9$ and the second by $8,$ we can then eliminate $y$ by adding the two equations: \[\frac{9x^2}{25} + y^2 + 8x = y^2 + 25,\]or \[9x^2 + 200x - 625=0.\]This quadratic factors as \[(9x-25)(x+25) = 0.\]Since $x = \tfrac{y^2}{8} + 2,$ it must be positive, so we have $x = \tfrac{25}{9}.$ Solving for $y$ in the equation $\tfrac{25}{9} = \tfrac{y^2}{8} + 2,$ we get $y = \pm \tfrac{2\sqrt{14}}{3}.$ Therefore, the distance between the two points is $2 \cdot \tfrac{2\sqrt{14}}{3} = \boxed{\tfrac{4\sqrt{14}}{3}}.$